the study of the logical and philosophical basis of mathematics, including whether the axioms of a given system ensure its completeness and its consistency. Because mathematics has served as a model for rational inquiry in the West and is used extensively in the sciences, foundational studies have far-reaching consequences for the reliability and extensibility of rational thought itself. For 2,000 years the foundations of mathematics seemed perfectly solid. Euclid ’s Elements (c. 300 bce), which presented a set of formal logical arguments based on a few basic terms and axioms, provided a systematic method of rational exploration that guided mathematicians, philosophers, and scientists well into the 19th century. Even serious objections to the lack of rigour in Sir Isaac Newton’s notion of fluxions (derivatives) in the calculus, raised by the Anglo-Irish empiricist George Berkeley (among others), did not call into question the basic foundations of mathematics. The discovery in the 19th...